1 Answer
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Differentiating both sides with respect to $h$ and applying the fundamental theorem of calculus, we obtain
$$f(h) = kf'(h)$$
This is a separable differential equation, and so its solution is given by
$$\int \frac{k}{f}df = \int dh$$
$$k\ln(f) = h+c$$
$$f(h) = ce^{h/k}$$
Since $f(0) = 0$, we have $c = 0$ and so $f(h) = 0$ for all $h$. This is the only solution.